The correct option for the formula for the energy (U) stored in a capacitor with capacitance (C) charged to a voltage (V) is: **B. U = 0.5CV²** ### Detailed Explanation 1. **Understanding Capacitance**: - A capacitor is a device that stores electrical energy in an electric field. The capacitance (C) of a capacitor is defined as the amount of charge (Q) it can store per unit voltage (V) applied across its plates. The relationship is given by the formula: \[ C = \frac{Q}{V} \] - Rearranging this gives us: \[ Q = CV \] 2. **Energy Stored in a Capacitor**: - The energy (U) stored in a capacitor can be derived from the work done to charge it. When charging a capacitor, the voltage across it increases as it accumulates charge. The work done (or energy stored) is the integral of the voltage with respect to charge: \[ U = \int_0^Q V \, dQ \] - Since \( V = \frac{Q}{C} \), we can substitute this into the integral: \[ U = \int_0^Q \frac{Q}{C} \, dQ \] - This integral evaluates to: \[ U = \frac{1}{C} \int_0^Q Q \, dQ = \frac{1}{C} \left[ \frac{Q^2}{2} \right]_0^Q = \frac{Q^2}{2C} \] - Now, substituting \( Q = CV \) into the equation gives: \[ U = \frac{(CV)^2}{2C} = \frac{C V^2}{2} \] - Thus, the formula for the energy stored in a capacitor is: \[ U = 0.5CV^2 \] ### Why Other Options Are Incorrect - **A. U = CV**: - This option suggests that the energy stored is equal to the product of capacitance and voltage. However, this does not account for the fact that the voltage increases as the capacitor charges. The energy stored is not simply the product of capacitance and voltage; it must consider the average voltage over the charging process. - **C. U = 2CV**: - This option incorrectly doubles the product of capacitance and voltage. The energy stored in a capacitor is not linear with respect to voltage; it is proportional to the square of the voltage, as shown in the correct formula. - **D. U = C/V**: - This option suggests that the energy stored is inversely proportional to voltage, which is incorrect. The energy stored increases with the square of the voltage, not decreases. This formula does not represent any physical relationship relevant to capacitors. ### Common Pitfalls - **Confusing Charge and Energy**: Students often confuse the charge stored in a capacitor (Q = CV) with the energy stored (U = 0.5CV²). It's important to remember that energy is related to the square of the voltage. - **Forgetting the Factor of 1/2**: When deriving the energy formula, it's easy to overlook the factor of 1/2 that arises from integrating the voltage over the charge. ### Revision Summary - The energy stored in a capacitor is given by the formula \( U = 0.5CV^2 \). - Capacitance (C) relates charge (Q) and voltage (V) through \( Q = CV \). - The energy is derived from the work done to charge the capacitor, considering the changing voltage. - Remember that energy is proportional to the square of the voltage, not linearly related to it.